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# If 0<x<1 which of the following inequalities must be

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Senior Manager
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If 0<x<1 which of the following inequalities must be [#permalink]

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27 Dec 2006, 23:13
This topic is locked. If you want to discuss this question please re-post it in the respective forum.

If 0<x<1 which of the following inequalities must be true?
I) x^5<x^3
II) x^4+x^5<x^3+x^2
III) x^4-x^5<x^2-x^3

a) NOne
b) I only
c) II only
d) I and II
e) I, II, and III
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27 Dec 2006, 23:38
(E) for me

I)
x < 1
=> x^5 < x^4 as x^4 > 0

Also,
=> x^4 < x^3 as x^3 > 0

So, x^5 < x^4 < x^3 >>>>> OK

II)
From the reasoning above, we have
x^5 < x^3. (A)

Similarly, we have x^4 < x^3 < x^2. That implies, x^4 < x^2. (B)

Adding the 2 inequalities (A) and (B), we obtain :
x^4+x^5 < x^3+x^2 >>>>> OK

III) Is x^4-x^5<x^2-x^3 true if 0 < x < 1 ?

x^4-x^5<x^2-x^3
<=> x^4*(1-x) < x^2*(1-x)
=> x^2*(1-x) < (1-x) if x != 0 and so x^2 > 0
=> x^2 < 1 if 1-x > 0 <=> x < 1

That means -1 < x < 0 OR 0 < x < 1.

Since 0 < x < 1, the inequality is true. >>>> OK.
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28 Dec 2006, 02:09
agree with fig. cannot explain it better.
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28 Dec 2006, 04:10
Sumithra wrote:
If 0<x<1 which of the following inequalities must be true?
I) x^5<x^3
II) x^4+x^5<x^3+x^2
III) x^4-x^5<x^2-x^3

a) NOne
b) I only
c) II only
d) I and II
e) I, II, and III

I) same as x3(x^2-1)<0 for 0<x<1, x^2 <1, x^3 >0 => I is true
II) same as: x^2(x^2-1) + x^3(x^2-1)<0
=>(x^2+x^3)(x^2-1)<0. again, we know x^2+x^3>0 and x^2-1<0 =>II is also true
III) same as: x^2(x^2-1) - x^3(x^2-1)<0
=> (x^2-x^3)(x^2-1)<0

for 0<x<1, x^2>x^3 => x^2-x^3 is positive and x^2-1<0 => III is also true.
Ans: E
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28 Dec 2006, 07:03

Thanks guys!
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29 Dec 2006, 04:37
E

choose x=0.1 and calculate the values.
29 Dec 2006, 04:37
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